How could a gap between array modifier created objects be closed?
Merge + First Last will not affect anything to the gap in your case. Because:
The reason that the first/last duplicate seem to be close is because of the Curve modifier, which affect AFTER the Array modifier.

You can try to first minimize the gap by adjusting in Array modifier, then go to the Edit mode of either the unit piece or the curve, S to scale, drag mouse while holding Shift to enable the very fine presision on scale, and zoom in the view, toggle Display modifier in Edit mode and Adjust edit cage to modifier result on all modifiers (the last two icons in every modifier panel). This would be helpful to see the result directly and simutaneously. Zoom in further, then scale again... repeat above steps (if necessary) until the gap is hardly noticed.

Example 1 (adjusting curve):



Example 2 (adjusting unit piece):



P.S.:


If you really want to perfectly remove that gap, you may have to apply all modifiers first, then go to Edit mode and remove doubles.
It is indeed possible to calculate exactly how much scaling is needed, but it will involve the math, and something about the relative position on pivots. I don't think it would be of much sense in most case.




[UPDATE]

Another way to get rid of the gap by Merge is using Object Offset feature:

Example 3 (using object offset):


About how to use it, I recommend to see this simple guide from CGCookie.com.